{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/understanding-the-cluster-lp-for-correlation","title":"Understanding the Cluster LP for Correlation Clustering","arxiv_id":"2404.17509","date":"2024-04-26","proceeding":null,"authors":["Nairen Cao","Vincent Cohen-Addad","Euiwoong Lee","Shi Li","Alantha Newman","Lukas Vogl"],"abstract":"In the classic Correlation Clustering problem introduced by Bansal, Blum, and Chawla (FOCS 2002), the input is a complete graph where edges are labeled either $+$ or $-$, and the goal is to find a partition of the vertices that minimizes the sum of the +edges across parts plus the sum of the -edges within parts. In recent years, Chawla, Makarychev, Schramm and Yaroslavtsev (STOC 2015) gave a 2.06-approximation by providing a near-optimal rounding of the standard LP, and Cohen-Addad, Lee, Li, and Newman (FOCS 2022, 2023) finally bypassed the integrality gap of 2 for this LP giving a $1.73$-approximation for the problem. In order to create a simple and unified framework for Correlation Clustering similar to those for typical approximate optimization tasks, we propose the cluster LP as a strong linear program for Correlation Clustering. We demonstrate the power of the cluster LP by presenting new rounding algorithms, and providing two analyses, one analytically proving a 1.56-approximation and the other solving a factor-revealing SDP to show a 1.485-approximation. Both proofs introduce principled methods by which to analyze the performance of the algorithm, resulting in a significantly improved approximation guarantee. Finally, we prove an integrality gap of $4/3$ for the cluster LP, showing our 1.485-upper bound cannot be drastically improved. Our gap instance directly inspires an improved NP-hardness of approximation with a ratio $24/23 \\approx 1.042$; no explicit hardness ratio was known before.","url_abs":"https://arxiv.org/abs/2404.17509v2","url_pdf":"https://arxiv.org/pdf/2404.17509v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"understanding-the-cluster-lp-for-correlation","repo_url":"https://github.com/correlationclusteringsdp/sdp1437code","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":{"status":"ok"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/2404.17509","atlas_url":"https://app.syntology.ai/?focus=2404.17509","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"2404.17509"}},"developers":"https://syntology.ai/developers","read_at":"2026-09-25T09:33:49+00:00","read_at_is":"when the build read Syntology's graph, not when any sample ran","claim":"Per-sample execution status on synthesized fixtures; not a correctness claim about the paper. Samples come from repositories linked to the paper, official or community; repo_kind says which.","repos":[{"provenance":"external:paperswithcode_snapshot_2025-07-28","url":"https://github.com/correlationclusteringsdp/sdp1437code","reach":{"status":"ok"}}],"summary":{"ran":5,"unverified":2},"by_repo_kind":{"official":{"samples":7,"ran":5,"repositories":1}},"repo_kind_vocabulary":{"official":"The archive marks this repository official for the paper","named_in_paper":"The archive records that the paper mentions this repository; it is not marked official","listed":"In the archive's code links for this paper, not marked official and not recorded as mentioned in the paper","found_in_text":"Syntology found this repository in the paper's own text; whether it is the authors' implementation is not asserted","community":"Not in the archive's code links for this paper; a community repository Syntology harvested"},"n_pointer_only_for_licence":7,"samples":[{"code_sha256_prefix":"3802afd1ba35cd97","entry":"filtersplittlingpoints","repo":"correlationclusteringsdp/sdp1437code","repo_kind":"official","path":"pure_cluster_lp/create_triangles.py","file_url":"https://github.com/correlationclusteringsdp/sdp1437code/blob/HEAD/pure_cluster_lp/create_triangles.py","link_basis":"first_harvest_node","language":"python","status":"ran","verification_level":1,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"NONE","inline_ok":false,"mcp_get_code":{"code_sha256":"3802afd1ba35cd97"}},{"code_sha256_prefix":"f0db2b8d92507a18","entry":"load_edgesIndex","repo":"correlationclusteringsdp/sdp1437code","repo_kind":"official","path":"pure_cluster_lp_extended_3_SA_server/run_create_triangles_server.py","file_url":"https://github.com/correlationclusteringsdp/sdp1437code/blob/HEAD/pure_cluster_lp_extended_3_SA_server/run_create_triangles_server.py","link_basis":"first_harvest_node","language":"python","status":"ran","verification_level":1,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":true,"licence":"NONE","inline_ok":false,"mcp_get_code":{"code_sha256":"f0db2b8d92507a18"}},{"code_sha256_prefix":"624d550a53d1efd2","entry":"psetEdgeAlgCost","repo":"correlationclusteringsdp/sdp1437code","repo_kind":"official","path":"pure_cluster_lp/create_triangles.py","file_url":"https://github.com/correlationclusteringsdp/sdp1437code/blob/HEAD/pure_cluster_lp/create_triangles.py","link_basis":"first_harvest_node","language":"python","status":"ran","verification_level":1,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":true,"licence":"NONE","inline_ok":false,"mcp_get_code":{"code_sha256":"624d550a53d1efd2"}},{"code_sha256_prefix":"60b3a82506e9b72f","entry":"roundIndex","repo":"correlationclusteringsdp/sdp1437code","repo_kind":"official","path":"pure_cluster_lp/create_triangles.py","file_url":"https://github.com/correlationclusteringsdp/sdp1437code/blob/HEAD/pure_cluster_lp/create_triangles.py","link_basis":"first_harvest_node","language":"python","status":"ran","verification_level":1,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":true,"licence":"NONE","inline_ok":false,"mcp_get_code":{"code_sha256":"60b3a82506e9b72f"}},{"code_sha256_prefix":"b1b0984f4b76e523","entry":"solve_sdp","repo":"correlationclusteringsdp/sdp1437code","repo_kind":"official","path":"pure_cluster_lp/sdp.py","file_url":"https://github.com/correlationclusteringsdp/sdp1437code/blob/HEAD/pure_cluster_lp/sdp.py","link_basis":"first_harvest_node","language":"python","status":"ran","verification_level":1,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"NONE","inline_ok":false,"mcp_get_code":{"code_sha256":"b1b0984f4b76e523"}},{"code_sha256_prefix":"5cbcb7ee2303971d","entry":"debug_createtriangles","repo":"correlationclusteringsdp/sdp1437code","repo_kind":"official","path":"pure_cluster_lp_extended_3_SA/debug_negative_ratio.py","file_url":"https://github.com/correlationclusteringsdp/sdp1437code/blob/HEAD/pure_cluster_lp_extended_3_SA/debug_negative_ratio.py","link_basis":"first_harvest_node","language":"python","status":"unverified","verification_level":0,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"NONE","inline_ok":false,"mcp_get_code":{"code_sha256":"5cbcb7ee2303971d"}},{"code_sha256_prefix":"a8ffbc7520ed8654","entry":"test_ratio","repo":"correlationclusteringsdp/sdp1437code","repo_kind":"official","path":"pure_cluster_lp_extended_3_SA/debug_negative_ratio.py","file_url":"https://github.com/correlationclusteringsdp/sdp1437code/blob/HEAD/pure_cluster_lp_extended_3_SA/debug_negative_ratio.py","link_basis":"first_harvest_node","language":"python","status":"unverified","verification_level":0,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"NONE","inline_ok":false,"mcp_get_code":{"code_sha256":"a8ffbc7520ed8654"}}]},"arxiv_metadata":null,"syntology_extracted_results":null}